Subgroups — #73068

Order 384.

127 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#41⋊C₃	C₂	1	1	yes	maximal, characteristic
2	SL(2,𝔽₃)⋊C₈	C₂	1	2	yes	maximal, complemented
3	#1476	—	3	1	no	maximal, Sylow
4	C₂×#41⋊C₃	C₄	1	1	yes	characteristic
5	C₄×SL(2,𝔽₃)	C₄	1	1	yes	characteristic
6	C₄×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₄×C₃⋊C₈	—	4	1	no	maximal
8	C₄×#41	D₃	1	1	yes	characteristic
9	#431	—	3	1	no	—
10	C₂×C₄×C₈	—	3	1	no	—
11	Q₈⋊C₈	—	3	2	no	—
12	D₄⋊C₈	—	3	2	no	—
13	#41⋊C₃	C₈	1	2	yes	complemented
14	C₂×SL(2,𝔽₃)	C₂×C₄	1	1	yes	characteristic
15	C₃×C₄²	—	4	1	no	—
16	C₂×C₃⋊C₈	—	4	2	no	—
17	C₂×#41	Dic₃	1	1	yes	characteristic
18	C₂²×C₈	—	6	1	no	—
19	C₄×Q₈	Dic₃	1	1	yes	characteristic
20	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
21	C₄×D₄	—	3	1	no	—
22	C₄×D₄	—	3	1	no	—
23	#117	—	3	1	no	—
24	C₂×C₄²	—	3	1	no	—
25	C₄⋊C₈	—	3	2	no	—
26	C₂²⋊C₈	—	6	1	no	—
27	C₄×C₈	—	3	2	no	—
28	C₄×C₈	—	6	1	no	—
29	C₄×C₈	—	6	1	no	—
30	C₂×C₃×C₄	—	4	1	no	—
31	C₂×C₃×C₄	—	4	1	no	—
32	C₂×C₃×C₄	—	4	1	no	—
33	SL(2,𝔽₃)	C₂×C₈	1	1	yes	characteristic, complemented
34	C₃⋊C₈	—	4	4	no	—
35	#41	—	3	2	no	—
36	#41	C₃⋊C₈	1	2	yes	complemented
37	C₂×Q₈	C₂×Dic₃	1	1	yes	characteristic
38	C₂×D₄	—	3	1	no	—
39	C₂²×C₄	—	3	1	no	—
40	C₂²×C₄	—	3	1	no	—
41	C₂²×C₄	—	3	1	no	—
42	C₂×C₈	—	6	2	no	—
43	C₂×C₈	—	6	4	no	—
44	C₂×C₈	—	6	2	no	—
45	C₄⋊C₄	—	3	1	no	—
46	C₄⋊C₄	—	3	1	no	—
47	C₂²⋊C₄	—	6	1	no	—
48	C₄²	—	3	1	no	—
49	C₄²	—	3	1	no	—
50	C₄²	—	3	1	no	—
51	C₄²	S₄	1	1	yes	characteristic
52	C₂²×C₃	—	4	1	no	—
53	C₃×C₄	—	4	2	no	—
54	C₃×C₄	—	4	2	no	—
55	C₃×C₄	—	4	2	no	—
56	C₂³	—	3	1	no	—
57	Q₈	—	3	1	no	—
58	Q₈	C₂×C₃⋊C₈	1	1	yes	characteristic
59	D₄	—	3	2	no	—
60	D₄	—	6	1	no	—
61	C₂×C₄	—	3	1	no	—
62	C₂×C₄	—	3	2	no	—
63	C₂×C₄	—	3	2	no	—
64	C₂×C₄	—	3	1	no	—
65	C₂×C₄	—	3	1	no	—
66	C₂×C₄	—	6	2	no	—
67	C₂×C₄	—	6	2	no	—
68	C₂×C₄	SL(2,ℤ/4)	1	1	yes	characteristic
69	C₂×C₄	SL(2,ℤ/4)	1	1	yes	characteristic
70	C₂×C₄	C₂×S₄	1	1	yes	characteristic
71	C₈	—	6	4	no	—
72	C₈	—	12	2	no	—
73	C₂×C₃	—	4	1	no	—
74	C₂×C₃	—	4	1	no	—
75	C₂×C₃	—	4	1	no	—
76	C₂²	—	3	2	no	—
77	C₂²	—	6	1	no	—
78	C₂²	—	6	1	no	—
79	C₂²	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
80	C₄	—	3	1	no	—
81	C₄	—	3	1	no	—
82	C₄	—	6	1	no	—
83	C₄	—	6	1	no	—
84	C₄	A₄⋊C₈	1	2	yes	—
85	C₄	GU(2,𝔽₃)	1	2	yes	—
86	C₄	#985	1	2	yes	—
87	C₃	—	4	1	no	Sylow
88	C₂	—	6	2	no	—
89	C₂	C₂×A₄⋊C₈	1	1	yes	characteristic
90	C₂	C₂×GU(2,𝔽₃)	1	1	yes	characteristic
91	C₂	#5424	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄²	S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂×C₈
Frattini	C₂×C₄	C₂×S₄
Fitting	C₄×#41	D₃
socle	C₂²	C₂×SL(2,ℤ/4)
prime radical	C₂×#41⋊C₃	C₄
cosocle	C₄×SL(2,𝔽₃)	C₂²
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. SL(2,𝔽₃)⋊C₈	C₂	semidirect
13. #41⋊C₃	C₈	semidirect
33. SL(2,𝔽₃)	C₂×C₈	semidirect
36. #41	C₃⋊C₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#41⋊C₃	2
SL(2,𝔽₃)⋊C₈	2
#1476	3
C₄×C₃⋊C₈	4